Unit disk graphs
TL;DR: It is shown that many standard graph theoretic problems remain NP-complete on unit disks, including coloring, independent set, domination, independent domination, and connected domination; NP-completeness for the domination problem is shown to hold even for grid graphs, a subclass of unit disk graphs.
About: This article is published in Discrete Mathematics.The article was published on 1991-01-02 and is currently open access. It has received 1525 citations till now. The article focuses on the topics: Indifference graph & Chordal graph.
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Book•
01 Jan 2006
TL;DR: This paper discusses Fixed-Parameter Algorithms, Parameterized Complexity Theory, and Selected Case Studies, and some of the techniques used in this work.
Abstract: PART I: FOUNDATIONS 1. Introduction to Fixed-Parameter Algorithms 2. Preliminaries and Agreements 3. Parameterized Complexity Theory - A Primer 4. Vertex Cover - An Illustrative Example 5. The Art of Problem Parameterization 6. Summary and Concluding Remarks PART II: ALGORITHMIC METHODS 7. Data Reduction and Problem Kernels 8. Depth-Bounded Search Trees 9. Dynamic Programming 10. Tree Decompositions of Graphs 11. Further Advanced Techniques 12. Summary and Concluding Remarks PART III: SOME THEORY, SOME CASE STUDIES 13. Parameterized Complexity Theory 14. Connections to Approximation Algorithms 15. Selected Case Studies 16. Zukunftsmusik References Index
1,730 citations
07 Nov 2002
TL;DR: This work presents their own distributed algorithm that outperforms the existing algorithms for minimum CDS and establishes the /spl Omega/(n log n) lower bound on the message complexity of any distributed algorithm for nontrivial CDS, which is thus message-optimal.
Abstract: The connected dominating set (CDS) has been proposed as the virtual backbone or spine of a wireless ad hoc network. Three distributed approximation algorithms have been proposed in the literature for minimum CDS. We first reinvestigate their performances. None of these algorithms have constant approximation factors. Thus these algorithms can not guarantee to generate a CDS of small size. Their message complexities can be as high as O(n/sup 2/), and their time complexities may also be as large as O(n/sup 2/) and O(n/sup 3/). We then present our own distributed algorithm that outperforms the existing algorithms. This algorithm has an approximation factor of at most 8, O(n) time complexity and O(n log n) message complexity. By establishing the /spl Omega/(n log n) lower bound on the message complexity of any distributed algorithm for nontrivial CDS, our algorithm is thus message-optimal.
834 citations
Cites background from "Unit disk graphs"
...However, finding an MCDS in unit-disk graphs is NP-hard [6], and thus only distributed approximation algorithms in polynomial time are practical for wireless ad hoc networks....
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...The topology of such wireless ad hoc network can be modeled as a unit-disk graph [6], a geometric graph in which there is an edge between two nodes if and only if their distance is at most one (see Figure 1)....
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TL;DR: This paper presents their own distributed algorithm that outperforms the existing algorithms for minimum CDS and establishes the Ω(nlog n) lower bound on the message complexity of any distributed algorithm for nontrivial CDS, thus message-optimal.
Abstract: Connected dominating set (CDS) has been proposed as virtual backbone or spine of wireless ad hoc networks. Three distributed approximation algorithms have been proposed in the literature for minimum CDS. In this paper, we first reinvestigate their performances. None of these algorithms have constant approximation factors. Thus these algorithms cannot guarantee to generate a CDS of small size. Their message complexities can be as high as O(n2), and their time complexities may also be as large as O(n2) and O(n3). We then present our own distributed algorithm that outperforms the existing algorithms. This algorithm has an approximation factor of at most 8, O(n) time complexity and O(n log n) message complexity. By establishing the Ω(n log n) lower bound on the message complexity of any distributed algorithm for nontrivial CDS, our algorithm is thus message-optimal.
652 citations
Cites background from "Unit disk graphs"
...However, finding an MCDS in unit-disk graphs is NP-hard [ 6 ], and thus only distributed approximation algorithms in polynomial time are practical for wireless ad hoc networks....
[...]
...The topology of such wireless ad hoc network can be modeled as a unit-disk graph [ 6 ], a geometric graph in which there is an edge between two nodes if and only if their distance is at most one (see figure 1)....
[...]
01 Jan 2001
TL;DR: This paper presents a clustering scheme to create a hierarchical control structure for multi-hop wireless networks and presents an efficient distributed implementation of the clustering algorithm for a set of wireless nodes to create the set of desired clusters.
Abstract: In this paper we present a clustering scheme to create a hierarchical control structure for multi-hop wireless networks. A cluster is defined as a subset of vertices, whose induced graph is connected. In addition, a cluster is required to obey certain constraints that are useful for management and scalability of the hierarchy. All these constraints cannot be met simultaneously for general graphs, but we show how such a clustering can be obtained for wireless network topologies. Finally, we present an efficient distributed implementation of our clustering algorithm for a set of wireless nodes to create the set of desired clusters.
616 citations
Cites background from "Unit disk graphs"
...In this case, the underlying communication graph, is a Unit Disk graph – defined in [20], [21] in terms of “distance” or “proximity” models, which consist of...
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TL;DR: This paper proposes a dominant pruning rule (called Rule k) that is more effective in reducing the dominating set derived from the marking process than the combination of Rules 1 and 2 and, surprisingly, in a restricted implementation with local neighborhood information, Rule k has the same communication complexity and less computation complexity.
Abstract: Efficient routing among a set of mobile hosts is one of the most important functions in ad hoc wireless networks. Routing based on a connected dominating set is a promising approach, where the search space for a route is reduced to the hosts in the set. A set is dominating if all the hosts in the system are either in the set or neighbors of hosts in the set. The efficiency of dominating-set-based routing mainly depends on the overhead introduced in the formation of the dominating set and the size of the dominating set. In this paper, we first review a localized formation of a connected dominating set called marking process and dominating-set-based routing. Then, we propose a dominant pruning rule to reduce the size of the dominating set. This dominant pruning rule (called Rule k) is a generalization of two existing rules (called Rule 1 and Rule 2, respectively). We prove that the vertex set derived by applying Rule k is still a connected dominating set. Rule k is more effective in reducing the dominating set derived from the marking process than the combination of Rules 1 and 2 and, surprisingly, in a restricted implementation with local neighborhood information, Rule k has the same communication complexity and less computation complexity. Simulation results confirm that Rule k outperforms Rules 1 and 2, especially in networks with relatively high vertex degree and high percentage of unidirectional links. We also prove that an upper bound exists on the average size of the dominating set derived from Rule k in its restricted implementation.
533 citations
Cites background from "Unit disk graphs"
...based on a connected dominating set is a promising approach, where the search space for a route is reduced to the hosts in the set....
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References
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01 Jan 1979
42,654 citations
Book•
01 Jan 1979
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
Abstract: This is the second edition of a quarterly column the purpose of which is to provide a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’’ W. H. Freeman & Co., San Francisco, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed. Readers having results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C355, Bell Laboratories, Murray Hill, NJ 07974, including details, or at least sketches, of any new proofs (full papers are preferred). In the case of unpublished results, please state explicitly that you would like the results mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this journal.
40,020 citations
01 Jan 1972
TL;DR: Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held, which made me aware of the importance of distinction between polynomial-time and superpolynomial-time solvability.
Abstract: Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held. These experiences made me aware that seemingly simple discrete optimization problems could hold the seeds of combinatorial explosions. The work of Dantzig, Fulkerson, Hoffman, Edmonds, Lawler and other pioneers on network flows, matching and matroids acquainted me with the elegant and efficient algorithms that were sometimes possible. Jack Edmonds’ papers and a few key discussions with him drew my attention to the crucial distinction between polynomial-time and superpolynomial-time solvability. I was also influenced by Jack’s emphasis on min-max theorems as a tool for fast verification of optimal solutions, which foreshadowed Steve Cook’s definition of the complexity class NP. Another influence was George Dantzig’s suggestion that integer programming could serve as a universal format for combinatorial optimization problems.
7,714 citations
Book•
01 Jan 1980
TL;DR: This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems and remains a stepping stone from which the reader may embark on one of many fascinating research trails.
Abstract: Algorithmic Graph Theory and Perfect Graphs, first published in 1980, has become the classic introduction to the field. This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems. It remains a stepping stone from which the reader may embark on one of many fascinating research trails. The past twenty years have been an amazingly fruitful period of research in algorithmic graph theory and structured families of graphs. Especially important have been the theory and applications of new intersection graph models such as generalizations of permutation graphs and interval graphs. These have lead to new families of perfect graphs and many algorithmic results. These are surveyed in the new Epilogue chapter in this second edition. New edition of the "Classic" book on the topic Wonderful introduction to a rich research area Leading author in the field of algorithmic graph theory Beautifully written for the new mathematician or computer scientist Comprehensive treatment
4,090 citations
TL;DR: This paper shows that a number of NP - complete problems remain NP -complete even when their domains are substantially restricted, and determines essentially the lowest possible upper bounds on node degree for which the problems remainNP -complete.
2,200 citations